Coconote
AI notes
AI voice & video notes
Export note
Try for free
Exploring Pi through Collisions
Aug 5, 2024
Mathematical Croquet: Exploring Collisions and Pi
Introduction
The lecture discusses a scenario involving two sliding blocks and a wall.
Assumptions:
No friction
Perfectly elastic collisions (no energy loss)
Aim: Count the number of collisions ("clacks") that occur in various scenarios.
Basic Scenario
Same Mass Blocks:
First block hits second block:
Transfers all momentum to the second block.
Second block bounces off the wall:
Transfers momentum back to the first block.
Total Clacks:
3
Different Mass Ratios
Case 1: First block 100 times the mass of the second:
Total collisions:
31
Case 2: First block 10,000 times the mass of the second:
Total collisions:
314
Case 3: First block 1,000,000 times the mass of the second:
Total collisions:
3,141
Pattern Identification
Observed pattern: When the mass of the first block is a power of 100 times the second, the number of collisions corresponds to the digits of pi.
Discovery Credit:
Originally discovered by mathematician Gregory Galperin in 1995, published in 2003.
Mentioned by viewer Henry Cavill.
Algorithm for Computing Pi
Steps to compute digits of pi using this phenomenon:
Implement a physics engine.
Choose the number of digits (d) of pi to compute.
Set mass of first block to be 100^(d-1).
Count collisions.
Example calculation for 20 digits:
Large block's mass = 100 billion billion billion billion times the mass of the smaller block (1 kg).
This results in counting
31 billion billion collisions
.
Clack frequency:
100 billion billion billion billion clacks per second
.
Implications and Observations
Theoretical implementation yields an impractical approach to computing pi.
Highlights the inefficiency of the method despite its elegance.
Raises the question: Why does pi appear in this context?
Pi is usually associated with circular geometry, hinting at a hidden circle in the problem.
The connection to conservation of energy will be explored in the next video.
Conclusion
Encouragement to explore the problem collaboratively.
Excitement for the next discussion about the underlying principles involving pi.
Thank you for watching!
📄
Full transcript